Strain rate, temperature and structure effects on the flow ...web.eng.ucsd.edu/mae/groups/markenscoff/symi_2012/presentations/... · of the flow stress with increasing temperature

G.I. Kanel, S.V. Razorenov, E.B. Zaretsky and S.I. Ashitkov

Joint Institute for High Temperatures of Russian Academy of Sciences, Moscow;Institute of Problems of Chemical Physics of Russian Academy of Sciences,

Chernogolovka,Department of Mechanical Engineering, Ben Gurion University, Beer Sheva, Israel

Strain rate, temperature and structure effects on the flow and fracture stresses of metals and alloys under shock-

wave loading

International Symposium on Current Problems in Solid Mechanics in honor of Professor R. J. Clifton

Brown UniversityGreek island of Symi, June 24-29, 2012

Motivation

• The need of the basis for developing the models and constitutiverelationships which would be workable over wide range of strain rates, stresses and temperatures.

• The need of new information about basic mechanisms of plastic deformation and fracture at high strain rate.

• Approaching the ultimate values of the shear strength and tensile strength.

• Coupling with atomistic simulations.

Strain rate and temperature effectsin general

• For low loading rates, plastic deformation is aided by thermal fluctuations. Dislocation motion is impeded at barriers and a combination of thermal agitation and applied stress is required to activate dislocations over the obstacles.

• In order to provide high rate of deformation one should apply so high stresses that are enough to overcome the usual dislocation barriers without any aid from thermal fluctuations.

• Under the high-rate conditions, viscous phonon drag becomes dominant in resistance to motion of dislocations. Since the phonon drag is proportional to the temperature, an increase of the flow stress with increasing temperature should occur at the highest strain rates.

Ideal shear strength

Shoc

k W

aves

Quasistatic, Hopkinson bars

Phonon drag controlled flow

Thermally activated flow

T2 > T1

T1

Flow

Stre

ss

Strain Rate

Frenkel’s estimation of the ideal shear strength:

τc ≈ G/2π (1926)

APPEARANCE OF MATERIAL PROPERTIES IN A FREE SURFACE VELOCITY HISTORY

Available loading conditions:

•Peak stresses: 0.1 − 100 GPa•Load durations: 10 ns −10 µs•Time resolution of the measurements:

≈ 1 ns

Hugoniot elastic limit: HEL = ρ0clufsHEL/2

Yield stress: Y = HEL (1-2ν)/(1-ν)

Spall strength: σsp = ρ0cb(∆ufs + δ)/2

Phase transition pressure:

pα→ε= ρ0D1ufs1/2

Sample

VISAR ufs(t)

Flyer plate

0.0 0.2 0.4 0.6 0.8 1.00.0

0.4

0.8

1.2Steel 40Kh

Spallsignal

∆ufs

Polymorphous transformation

Hugoniot elastic limit (HEL)

Free

Sur

face

Vel

ocity

, km

/s

Time (µsec)

Attenuators

Delay line

SHGTarget

Ti:S laser: τL=150 fs, λ=800 nm

CCD

Pump: 800nm, 150 fs

Probe: 400nm, 150 fs

Interferometer

Femtosecond interferometric microscopy for study laser-driven shock waves

Temporal resolution: 10-13 sSpatial resolution: 2 µmPhase measurement accuracy: ~ π/200 (1nm)

Initialbefore pump pulse

Transient1 ns delay after pump

Free surface displacement histories

0 50 100 150 2000

20

40

60

80 Al

1200 nm

760 nm

500 nm

Dis

plac

emen

t z,

nm

Time t, ps

Development of Elastic Precursor Waves in Relaxing Materials

0 50 1000.0

0.2

0.4

0.6

0.8

1.0

0 200 400 600 800 1000

0.25 mm

1.05 mm

0.44 mm

Free

Sur

face

Vel

ocity

, km

/s

Time, ns

Mg - 4.5% Al alloy

HEL

1.9

5 mm

10.8 mm

tu

ctV

dtdu

tctu

dhd

l

x

l

x

∂∂

+∂∂

=

∂∂

+∂∂

−=

1

1

0HEL

0HEL

ρ

σρσ

( )Vccdh

dl

&&l22

0HEL

12 ρσσ +=

• G.E. Duvall. In: Stress Waves in Inelastic Solids, edited by H. Kolsky and W. Prager, 1964

• Ahrens T.J. and Duvall G.E. J. Geophys. Res., 71(18), 4349-4360 (1966).

• J. R. Asay, G. R. Fowles, and Y. Gupta, J. Appl. Phys. 43, 744 (1972).

l

px

cG

dhd γσ &

34

HEL

−=

x

σHEL

Decay of elastic precursor wave in aluminum

10-3 10-2 10-1 100 1010.01

0.1

1

10Ashitkov et al, 2010

Al

σHEL= 0.16/X0.63

Whitley et al, 2011

Gupta et al, 2009

Ashitkov et al,2011

Garkushin et al, 2010

Arvidsson et al, 1975

Winey et al, 2009

(G0 /G

)σH

EL, G

Pa

Distance, mm

The data are well described by empirical relationship ασ −= )( 0hhSHEL

h0 = 1 mm, S = 0.16 GPa, α = 0.63

l

p

cG

dhd γσ &

34

HEL

−=GhhhcS l

p0

)1(0 )(

43 +−

=ααγ& Gh

cSSG

E lp

0

1

43 ατγ

αα+

⎟⎟⎠

⎞⎜⎜⎝

⎛ ′=&

Experiments withFemtosecond Laser Pulses

Plate impact

}

}

AtomisticSimulations:HEL up to15-20 GPa

Relationship between the plastic strain rate and the shear stress at HEL

10-2 10-1 100 101102

103

104

105

106

107

108

109

1010

γ '= 9.1 x 107 τ 2.59

1 µm

10 mm

Al

Plas

tic s

train

rate

γ', s

-1

(G0 /G)τHEL, GPa

At ~(2-5)×103 s-1 the Hopkinson bar tests show sharp increase of sensitivity of the flow stress to strain rate (see K. Sakino. J. Phys. IV France, 10, Pr9-57 – 62, 2000, as an example).Decay of HEL sharply decelerates at the distance of 5-10 mm (~103 s-1 strain rate)

101 102 103 104 105 106 1070

50

100

150

200

250

300

τ = 10 + 2 lg(γ ')τ HEL at 5 -10 mm

AlτHEL= (γ '/9.1x107)0.39

τ, M

Pa

Plastic strain rate γ ', s-1

Strain rate and stress in plastic shock wave

0 100 200 300 4000.0

0.2

0.4

0.6

0.8

dγmax/dt = (dεx/dt )/2 = (0.4-0.6)x107 s-1

dεx/dt = (dufs/dt )/2US = (0.8-1.2)x107 s-1

dufs/dt = (0.8-1.2)x108 km/s2

Mg-4.5%Al, 10.8 mm

Free

Sur

face

Vel

ocity

, km

/s

Time, ns

0.00 0.02 0.04 0.06 0.080

50

100

150

200 τ hmax = 3/4 (σx-p) = 139.2 MPa

σ x - p

, MPa

1-V/V0

• L.C.Chhabildas, and J.R. Asay. J. Appl. Phys., 50(4), 2749-2756 (1979)

• J.W. Swegle and D.E. Grady. J. Appl. Phys. 58, 692 (1985)• D. E. Grady. Structured shock waves and the fourth-power law.

J. Appl. Phys. 107, 013506 (2010)

(4/3)τ

(2/3)Y

HEL

Hydrostat p(v)

Rayleigh line

Hugoniot σ

x(v)

Pres

sure

, Stre

ss

Specific Volume

Spx Uu&& =εTotal strain rate in steady wave:

2xεγ && =Shear strain rate at 1-D compression:

Gx

p 22τεγ&&

& −=Plastic shear strain rate:

Accelerated relaxation in plastic shock wave

• The strain rate in plastic shock wave is faster that at the HEL by an order of magnitude.

• Acceleration of plastic deformation is associated with multiplication of dislocations.

103 104 105 106 107

50

100

150

200

250

300

γ ' ~ τ 2

Mg-4.5%Al

10 mm

0.25 mm

In plastic shockat 5-10 mm

At HELτ = 11γ ' 0.24

τ, M

Pa

γ'p, s-1

Temperature effects

SHOCK-WAVE LOADING OF ALUMINUM SINGLE CRYSTALS AT ELEVATED TEMPERATURES

• The Hugoniot elastic limit and the rise time of plastic shock wave unexpectedly grow with increasing the temperature;

• A strong rate sensitivity results in a strong decay of the elastic precursor wave.

0.0 0.1 0.2 0.30

100

200

300

400

500

600

700

Sample thickness 2.85 mm.

407oC20oC

650oC

Free

Sur

face

Vel

ocity

, m/s

Time, µs0 20 40 60 80

0

200

400

600

800

260µm, 622oC

425 µm, 20oCFree

Sur

face

Vel

ocity

, m/s

Time, ns

Behavior of AD1 aluminum at ~600°C

0 50 100 150 2000

200

400

600AD1

5.37 mm20 oC

5.18 mm605 oC

Free

Sur

face

Vel

ocity

, m/s

Time, ns

0 100 200 300 400 5000

200

400

600

800

4

32

1

3 - 5.18 mm, 605oC

1 - 0.595 mm, 592oC

4 - 10.01 mm, 612oC

2 - 2.23 mm, 619oC

Free

Sur

face

Vel

ocity

, m/s

Time, ns

Increase in the HEL and rise time of plastic shock wave as compared to the room temperature data.

Initial flow stress at high strain rates and elevated temperatures

0.01 0.1 1 100.01

0.1

1

Al AD1

20oCσHEL= 0.625 / x

0.362600oC

σ HEL

, GPa

Distance, mm

101 102 103 104 105 106 10710

20

40

6080

100

200

400

Al AD1

20oC

600oC

She

ar s

tress

, MP

a

Plastic strain rate, s-1

Ultimate shear strength (“ideal strength”) should decrease with increasing the temperature.

Strain rate and temperature effectsin aluminum

Shoc

k W

aves

Quasistatic, Hopkinson bars

Phonon drag controlled flow

Thermally activated flow

T2 > T1

T1

Flow

Stre

ss

Strain Rate

The shock-wave tests confirm general view on strain rate and temperature effects:• The transitions in rate controlling mechanism occur at ~3×103 s-1 when the

temperature is 20 °C and at smaller strain rate ~102 s-1 at 600 °C;• Anomalous thermal hardening takes place in the high rate sensitivity region.

101 102 103 104 105 106 10710

20

40

6080

100

200

400

Al AD1

20oC

600oC

She

ar s

tress

, MP

aPlastic strain rate, s-1

Strain rate and temperature effectsin pure aluminum

103 104 105 106 1070

100

200

300

400

500

Al 99.99%

1+= α

α

γτ &ALinear

2 mm

0.1 mm

HELPlasticshock wave

932 K

300 K

932 K

800 K

500 K

300 K

τ = 10 + 2 lg(γ ')

She

ar s

tress

, MPa

Plastic strain rate, s-1

The linear dependences τ (T ) at fixed strain rates are in agreement with the mechanism of control of the dislocations motion by phonon friction.

200 400 600 800 10000

50

100

150

200

250

300

350

106 s-1

105 s-1

104 s-1

She

ar S

tress

, MP

a

Temperature, K

Relationship between the plastic strain rate and the shear stress at HEL

• The motion of dislocations sharply decelerates at ~100 MPa of the shear stress that is not compatible with the observed strong dependence γ’(τ).

• The dislocation density of 1015 m-2 is required to provide 109 s-1 strain rate with a sound-speed dislocation motion.

• Probably the process is mainly controlled by the rate of nucleation of dislocations (or shear bands).

Velocity as a function of the applied shear stress for an edge dislocation in aluminum.

D.L. Olmsted, .G. Hecto, W.A. Curtin and R.J. Clifton. Modelling Simul. Mater. Sci. Eng. 13, 371 (2005)

103

104

105

106

107

10 20 40 60 80100 200 400

2.7 < β < 4

βτγ B=&

Al 99.99%

2 mm

0.1 mm

932 K

800 K

500 K

300 K

Shear stress, MPa

Plas

tic s

train

rate

, s-1

Comparison of shock response of Al single crystal, polycrystalline Al and commercial aluminum AD1

0.0 0.1 0.2 0.30

100

200

300

400

500

600

700

Al

Polycrystalline Al 99.99630oC 2.0 mm

AD1 2 mm, 619 oC

Single crystal622oC 2.97 mm

Free

Sur

face

Vel

ocity

, m/s

Time, µs

The spike-like shape of the elastic precursor wave is an evidence of accelerating stress relaxation as a result of multiplication of dislocations behind the precursor front.In the alloys the large plastic strain rate immediately behind the elastic precursor front is provided by an abundance of the dislocation nucleation sites without essential contribution of the multiplication processes. It may be supposed that increase of the temperature does not ease the nucleation and multiplication processes.

0.1 1

0.1

1

Al, average

851 K

Al 6061

Pure Al

HE

L, G

Pa

Distance, mm

D16 aluminum alloy (Al 2024)

0 50 100 150 2000

200

400

600

800

D16 (Al 2024)

0.5 mm5 mm

Free

Sur

face

Vel

ocity

, m/s

Time, ns

Weak precursor decayThe HEL of the alloy exceeds that of aluminum at the propagation distances more than 0.1 mm

10-3 10-2 10-1 100 1010.01

0.1

1

10

D16 (Al 2024)

Ashitkov et al, 2010

Al

σHEL= 0.16/X0.63

Whitley et al, 2011

Gupta et al, 2009

Ashitkov et al,2011

Garkushin et al, 2010

Arvidsson et al, 1975

Winey et al, 2009

(G0 /G

)σH

EL, G

PaDistance, mm

Shock behavior of iron at normal and elevated temperatures

0 100 200 300 400 500 6000.0

0.5

1.0

1.5

2.0

Y

HEL

Yiel

d S

tress

, GP

a

Temperature, oC

• Weak precursor decay• The yield stress decreases monotonously with heating

10-4 10-3 10-2 10-1 100 101

R.F. Smith et al, 2011

30

20

1086

4

2

1

HEL

Hug

onio

t Ela

stic

Lim

it, G

Pa

Distance, mm

YIELD STRENGTH OF TITANIUMS AT ELEVATED TEMPERATURES

-200 0 200 400 6000.0

0.5

1.0

1.5

2.0

Commercial Ti

Ti 99.99%

Ti-6-22-22S

Yie

ld S

tress

, GP

a

Temperature, oC

• The flow stress in the pure metal is small and comparable with the phonon friction forces. Therefore the growth of the latter contributes essentially into the drag of the dislocations.

• Alloys contain numerous obstacles that have been created specifically to increase the yield strength. The stress needed to overcome these obstacles far exceeds the forces of phonon drag.

10-6 10-4 10-2 100 102 104 1060.0

0.5

1.0

1.5

2.0

Ti6-22-22S

1-D strain (Shock)

Uniaxial stress

Yiel

d St

ress

, Y0.

2, G

Pa

Strain Rate, s-1

Temperature effect at large strain

S. E. Grunschel, R. J. Clifton, and T. Jiao. In: Shock compression of Condense Matter 2011, p.1335

The work hardening may cause larger effect than the phonon friction

103 104 105 106 1070

100

200

300

400

500

Al 99.99%

1+= α

α

γτ &ALinear

2 mm

0.1 mm

HELPlasticshock wave

932 K

300 K

932 K

800 K

500 K

300 K

τ = 10 + 2 lg(γ ')

She

ar s

tress

, MPa

Plastic strain rate, s-1

Data for plastic strain γp = 0 at the HELand γp ≈ 0.01 in plastic shock wave

Grain size effects

Armco iron

50 µm

50 µm

0 200 400 6000

50

100

150

200

250

300Armco iron, 2.7 mm

Forged

Fre

e Su

rface

Vel

ocity

, m/s

Time, ns

As received

0 25 50 750

100

200

300

400

500

600

Iron 0.2 mm thick

Armco, forged

Armco, as-received

Free

Sur

face

Vel

ocity

, m/s

Time, ns

Refinement of the grain structure results in twofold increase of and some smaller increase of the dynamic yield strength.

Coarse-grained and ultrafine-grained tantalum

0.00 0.05 0.10 0.15 0.200

100

200

300

400

500

0.00 0.05 0.10 0.15 0.20

CG, HR = 77 UFG, HR = 104

Free

sur

face

vel

ocity

, m/s

Time, µs

HEL

Distance

• In spite of larger hardness, ultrafine-grained tantalum demonstrates lower HEL and faster plastic compression.

• Obviously, in the UFG material the elastic precursor wave decays faster.

• Grain boundaries are the dislocation sources.

0.0 0.5 1.0 1.5 2.0 2.5

1.5

2.0

2.5

3.0

3.5

UFG

CG

HEL

, GPa

Sample Thickness, mm

Conclusions for development of material models

Strain Rate

Flo

w S

tress

High Temperature

Hardened

Anomalous thermal hardening occurs in the domain of high rate sensitivity of the flow stress;Mechanical hardening of a material shifts the transition towards higher strain rates; Hardened material may demonstrate less flow stress at high strain rate. Both the work hardening and the work softening may accompany the high-rate deformation

Spall fracture

SPALL PHENOMENA UNDER SHOCK LOADING

Spalling is the process of internal rupture of a body due to tensile stresses generated as a result of a compression pulse reflected from the free surface.

σx

σx

Freesurface

x

Freesurface

x

The wave dynamics at reflection of a shock pulse from the free surface

∆ufs

uf

um

u0

ufs

t0

K

umu0

(tail)SpallStrength

0

C-

C+

C+

Riemann'sIsentropes

Hugoniot

u

p

Acoustic approach: fsoosp uc ∆ρ=σ21

The peak tensile stress just before the fracture corresponds to the intersection point K of the Riemann’s isentropes

t

x

K

C_

C+Sp

all

Free surface

Shock front

0

SPALL STRENGTH OF ALUMINUMS OF DIFFERENT PURITY

• Single crystals always exhibit highest dynamic tensile strength.• Pure metals often show higher spall strength than harder alloys.

0.0 0.1 0.2 0.3 0.4 0.50

200

400

600

Al 99.99 %

Commercial aluminum AD1(99.3% purity)

Al single crystal

Free

Sur

face

Vel

ocity

, m/s

Time, µs

( )uuc pbbsp δρσ +∆= 021

104 105 106 107

σ sp / σ id

WaterPMMA

Epoxy

0.04

0.06

0.08

0.50.4

0.3

0.2

0.1

CuAl

Fe

Mo

.V/V0, s

-1

APPROACHING THE IDEAL STRENGTH

• As much as 30 % of ideal strength is reached at load duration of a nanosecond range.

0,6 0,8 1,0 1,2-60

-40

-20

0

20

Water

PMMA

Mo

Fe

Cu

Al

Tension Compression

Pres

sure

, GPa

ρ/ρ0

Equations of State in the Negative Pressure Domain

0.21 – EOS (Speedy, 1982)0.22 – MD (Netz et al, 2001)0.28Water

1.34Epoxy1.39PMMA

13.5 – EOS (Faizullin and Skripov, 2007)

27.9 (Friakyz et al, 2003)27.7 (Černý and Pokluda, 2007)13.4 (Sin’ko and Smirnov, 2004)

31.6Iron

41.6 (Joshi and Gupta, 2007)42.9 (Černý and Pokluda, 2007)55Molybdenum

21.0 – EOS (Faizullin and Skripov, 2007)19.8 (Černý and Pokluda, 2007)23.3Copper

12.2 – EOS (Faizullin and Skripov, 2007)11.7 (Sin’ko and Smirnov, 2002)13.4Aluminum

σid at 300 K from EOS or MD, GPaσid ab initio, GPaEstimated

σid, GPaMatter

APPROACHING THE IDEAL STRENGTH

• At least 60 % of ideal strength is reached at load duration of apicosecond range.

103 104 105 106 107 108 109 10100.4

0.60.8

1

2

4

68

10

20

Al

FoilsMoshe, Eliezer, 2000

Ashitkov et al, 2010

MDσspall = 0.4 + 3.75(ε' /108 )0.2

Ideal strength

Plate impacts Laser & particle beams

Al 99.99%

Single Crystals

AD1

Spal

l stre

ngth

, GPa

Strain rate, s-1

Intragranular and intergranular fracture of Cu + 0.1%Si

Grain size effect on the spall strength.Armco iron and tantalum.

0 25 50 750

100

200

300

400

500

600

σsp

= 5.25 GPaSp.thick. 40 µm

σsp

= 3.62 GPaSp.thick. 33 µm

σsp

=6.80 GPaSp.thick. 43 µm

Iron 0.2 mm thick

Fe single crystalArmco, forged

Armco, as-received

Free

Sur

face

Vel

ocity

, m/s

Time, ns

0 200 400 6000

50

100

150

200

250

300

Single crystal, σsp = 4 GPa

Armco iron, 2.7 mm

Forgedσsp= 3.3 GPa

Fre

e Su

rface

Vel

ocity

, m/s

Time, ns

As received,σsp= 2.1 GPa

• Refinement of the grain structure results in increase of the spall strength.

• The spall strength does not correlate with the HEL.

0.00 0.05 0.10 0.15 0.200

100

200

300

400

500

0.00 0.05 0.10 0.15 0.20

Ta

CG, HR = 77 UFG, HR = 104

Free

sur

face

vel

ocity

, m/s

Time, µs

Ta

Influence of a heat treatment and polymorphic transformation on the spall behavior of steel

0.0 0.4 0.8 1.20.0

0.4

0.8

1.2

As received

Quenched

Free

Sur

face

Vel

ocity

, km

/sec

Time, µsec

Chromium-doped structural steel. Chemical composition:C − 0.4 wt%, Si − 0.3 wt%, Mn − 0.6 wt%, Cr − 1 wt%

4.75~80Quenched

4.0~75As received

4.720Quenched

2.919*As received

4.27.1Quenched

2.37.1As received

Spall Strength,

GPa

Peak Stress, GPa

State of material

* Spall fracture occurred at the boundary between the initial and transformed matter in this shot

Refinement of the grain size as a result of reversible α-ε transformation increases the spall strength value

SPALL STRENGTH OF SINGLE CRYSTALS AND POLYCRYSTALLINE METALS AT MELTING

• The strength of polycrystalline metals drops when the material begins to melt whereas single crystals maintain a high resistance to spall fracture when melting should start;

• In polycrystalline solids melting may start along grain boundaries at temperatures below the melting temperature of the crystal: pre-melting phenomenon;

• Superheated solid states were realized in the crystals under tension

0 100 200 300 400 500 600 7000

1

2

3

4

Zn (0001)

3.106s-1

5.105s-1

Al

Spal

l Stre

ngth

, GPa

Temperature, oC0,4 0,6 0,8 1,0

0,0

0,5

1,0

1,5

MgAl

Spal

l Stre

ngth

, GPa

- Magnesium Mg95- Aluminum AD1

Homologous Temperature T/Tm

Thank you for your attention!